Electrodynamics

Electrodynamics Introduction: The basic relationships of the electrostatics and the steady magnetic fields were dealt with so far, represented by E(x, y, z) and H(x, y, z). We have restricted our discussion to static or time – invariant EM fields. Now, we shall examine situations in which electric and magnetic fields are dynamic or time – varying. It should be noted that in static EM fields, electric and magnetic fields are independent of each other, whereas, in dynamic EM fields, the two fields are interdependent. Electrostatic fields are usually produced by static electric charges, whereas, magnetostatic fields are due to motion of electric charges with uniform velocity (direct current) or static magnetic charges (magnetic poles); time – varying fields or waves are usually due to accelerated charges or time – varying currents such as sinusoidal or rectangular or triangular. Two new concepts will be introduced; the electric field produced by a changing magnetic field and the magnetic field produced by changing electric field. As a result of these concepts, Maxwell’s equations expressed so far and the boundary conditions for static EM fields will be modified to account for the time variation of the fields. Faraday’s Law: After Oersted demonstrated in 1820 that an electric current affected a compass needle, Faraday professed his belief that if a current could produce a magnetic field, then a magnetic field should be able to produce a current. In 1831, about 11 years after Oersted’s discovery Faraday in London and Henry in New York discovered that a time – varying magnetic field would produce an electric current. According to Faraday’s experiments, a static magnetic field produces no current flow, but a time – varying field produces an induced voltage in a closed circuit, which causes a current flow. This induced voltage is called electromotive force or simply emf. Faraday discovered that the induced emf, Vemf (in volts), in any closed circuit is equal to the time rate of change of the magnetic flux linkage by the circuit. Faraday’s law can be stated customarily as

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Vemf = -

=-N

Where λ = Nφ, is the flux linkage, N is the number of turns in the circuit and Φ is the flux through each turn. The minus sign is an indication that the emf is in such a direction as to produce a current whose flux, if added to the original flux, would reduce the magnitude of the emf. This statement that the induced voltage acts to produce an opposing flux is known as Lenz’s law. Sources of emf include electric generators, batteries, thermocouples, fuel cells or photovoltaic cells, which all convert non electrical energy into electrical energy. Considering the figure, the electric circuit consists of the battery which is a source of emf. The electro chemical action of the battery results in an emf – produced field Ef. Due to the accumulation of charges at the battery terminals, an electrostatic field Ee (= -

) also exits. The

total electric field at any point is given by, E = Ef + Ee Ef is zero outside the battery, Ef and Ee have opposite directions in the battery, and the direction of Ee inside the battery is opposite to that outside it. Integrating the above equation over the closed circuit, we have, = L Where

f

. dL

L

= 0 because Ee is conservative.

The emf of the battery is the line integral of the emf – produced field, that is, Vemf =

f

. dL = -

e

. dL = IR

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Since Ef and Ee are equal but opposite within the battery. If any part of the path is changed, generally the emf changes. The departure from static results is clearly shown by the above equation for an electric field intensity resulting from a static charge distribution must lead to zero potential difference about the closed path. In electrostatics, the line integral leads to a potential difference; with time – varying fields, the result is an emf or voltage. It is important to note the following facts: An electrostatic field Ee cannot maintain a steady current in a closed circuit since

L An emf – produced field Ef is non conservative. Except in electrostatics, voltage and potential difference are usually not equivalent. Transforme r and motional electromotive forces: We shall now see how Faraday’s law links electric and magnetic fields. For a circuit with single turn (N=1), the above equation becomes, Vemf = -

Replacing φ in the above equation with the surface integral of B, we have, Vemf =

=L

∫B. dS S

where the fingers of our right hand indicate the direction of closed path and our thumb indicates the direction of dS. It is clear from the above equation that in a time – varying situation both electric and magnetic fields are present and are interrelated. A flux density B in the direction of dS and increasing with time thus produces an average value of E which is opposite to the positive direction about the closed path. The variation of flux with time as in equation (1) or equation (5) may be caused in three ways: By having a stationary loop in time – varying B field. By having a time – varying loop area in a static B field. By having a time – varying loop area in a time – varying B field. 3

Stationary loop in time – varying B field. (Transforme r emf) In the figure, a stationary loop is considered. The magnetic flux is the only time varying quantity on the right hand side of equation (5) and a partial derivative may be taken under the integral sign, =-∫

Vemf = L

.dS

S

This emf induced by the time – varying current (producing the time – varying B field) in a stationary loop is often referred to as transformer emf in power analysis, since it is due to transformer action. By applying Stoke’s theorem to the middle term in equation (6), we obtain, ∫

) . dS = - ∫

.dS

S S where the surface integrals may be taken over identical surfaces. The surfaces are perfectly general and may be chosen as differentials, =This is one of four Maxwell’s equations as written in differential or point form, the form in which they are most generally used for time – varying fields. It shows that the time – varying E field is not conservative (

≠ 0). This does not imply that the principles of energy

conservation are violated. The work done in taking a charge about a closed path in a time – varying electric field is due to the energy from the time – varying magnetic field. Moving loop in Static B field (Motional emf) When a conducting loop is moving in a static B field, an emf is induced in the loop. The force on a charge moving with uniform velocity u in a magnetic field B is Fm = Q u X B

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We define the motional electric field Em, as Em = Fm /Q = u X B If we consider a conducting loop, moving with uniform velocity u as consisting of a large number of free electrons, the emf induced in the loop is Vemf = ∮Em. dL = ∮(u X B). dL L L This type of emf is called motional emf or flux – cutting emf, because it is due to motional action. It is the kind of emf found in electrical machines such as motors, generators and alternators. u X B is the induced electric field intensity and its direction is the same as that of the induced current in the conductor. This fact enables us to establish the polarity of the motional emf across the two ends of a conductor. In a nutshell, the induced current due to the motional emf is in the direction of the induced electric field. Moving loop in time – varying field: In the general case, a moving conducting loop is in a time – varying magnetic field. Both transformer emf and motional emf are present. Therefore, the total emf is =-∫

Vemf = L

.dS + ∮(u X B). dL

S

L

or, =-

+

X (u X B)

Equation of continuity for time – varying fields: Since current is simply charge in motion, the total current flowing out of some volume must be equal to the rate of decrease of charge within the volume, assuming that charge cannot be created or destroyed. This concept is essential in understanding why current flows in the leads to a capacitor during charge or discharge when no current flows between the capacitor plates. The explanation is simply that the current flow is accompanied by a charge build up on the plates. In mathematical terms, this conservation of charge concept can be stated as

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∮J.dS = -

∫ρdV

If the region of integration is stationary, the above relation becomes, ∮J.dS = - ∫

dV

The divergence theorem now may be applied in order to change the surface integral into a volume integral.

∫

dV = - ∫

dV

If the above relation is to hold for any arbitrary volume, then it must be true that, =This is time – varying form of the equation of continuity. Inconsistency of Ampe re’s Law (or) Displace ment Curre nt: Faraday’s experimental law has been used to obtain one of Maxwell’s equations in differential form, =which shows us that a time – varying magnetic field produces an electric field. Remembering the definition of the curl, we see that this electric field has the special property of circulation; its line integral about a general closed path is not zero. Now, let us consider the case of the time – changing electric field. Considering the point form of Ampere’s circuital law as it applies to steady magnetic fields, =J and show its inadequacy for time – varying conditions by taking the divergence of each side,

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The divergence of curl is identically zero, so

is also zero. The continuity of current requires

that,

This is an unrealistic limitation and the equation (2) must be amended before we can accept it for time – varying fields. To do this, we add a term to equation (2) so that it becomes, = J + Jd Again, the divergence of the curl of any vector is zero. Hence, =0=

+ .Jd

In order the above equation to agree with equation (4), .Jd = -

=

=

(

=

.(

)

Jd = This is Maxwell’s equation (based on Ampere’s circuit law) for a time – varying field. The term

Jd =

is known as displacement current density and J is the conduction current density

(J = σE). Equation (9) has not been derived. It is merely a form we have obtained which does not disagree with the continuity equation. Without the term J d, the propagation of electromagnetic waves (radio or TV waves) would be impossible. At low frequencies, J d is usually neglected compared with J. However, at radio frequencies, the two terms are comparable. Based on displacement current density, we define the displacement current as Id = ∫ Jd . dS = ∫

.dS

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Displacement current is a result of time – varying electric field. A typical example of such current is the current through a capacitor when an alternating voltage source is applied to its plates. Integration of equation (9) over a surface and application of Stoke’s theorem leads to the integral form, ∮ H . dS = ∫(

+ J).dS

The above equation states that the magneto motive force around a closed path is equal to the total current enclosed by the path. Maxwell’s Equations: Maxwell’s equations are four of the most influential equations in science; Gauss’s law for electric fields, Gauss’s law for magnetic fields, Faraday’s law and Ampere – Maxwell law. If you need a testament to the power of Maxwell’s equations, look around yo u – radio, television, radar, wireless Internet access and Bluetooth technology are a few examples of contemporary technology rooted in electromagnetic field theory. Maxwell’s celebrated work led to the discovery of electromagnetic waves. Through his theoretical efforts, Maxwell published the first unified theory of electricity and magnetism. It introduced the concept of displacement current and predicted the existence of electromagnetic waves. Maxwell’s equations in generalized form in both point or differential form and integral forms are as listed below: Differential form = ρv

Integral form ∮D.dS = ∫ ρ v dV S V

∮B.dS = 0 S = - ∫ .dS

=0 ==J+

L S ∮ H . dL = ∫( + J).dS L

Remarks Gauss’s Law Gauss’s Law for magnetic fields Faraday’s Law Ampere’s Circuit Law

S

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Equation (1) essentially states that charge density is a source (or sink) of electric flux lines. Note that we can no longer say that all electric flux begins and terminates on charge, because the point form of Faraday’s law shows that E and hence, D, may have circulation if a changing magnetic field is present. Thus, the lines of electric flux may form closed loops. However, the converse is still true, and every coulomb of charge must have one coulomb of electric flux diverging from it. Equation (2) again acknowledges the fact that magnetic charges or monopoles are not known to exist. Magnetic flux is always found in closed loops and never diverges from a point source. Word statement of the field equations: A word statement of the significance of the field equations is readily obtained from their mathematical statement in the integral form. The total electric displacement through the surface enclosing a volume is equal to the total charge within the volume. The net magnetic flux emerging through any closed surface is zero. The electromotive force around a closed path is equal to the time der ivative of the magnetic displacement through any surface bounded by the path. The magneto motive force around a closed path is equal to the conduction current plus the time derivative of the electric displacement through any surface bounded by the path. The other equations that go hand in hand with Maxwell’s equations are: The Lorentz force equation: F = Q(E + u x B) is associated with Maxwell’s equations.

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Also the equation of continuity, =is implicit in Maxwell’s equations. The concepts of linearity, isotropy and homogeneity of a material medium still apply for time – varying fields; in a linear, homogeneous and isotropic medium characterized by σ,ε and μ, the constitutive relations D = ε E = ε0 E + P B = μH = μ0 (H + M) J = σE + ρV u hold for time – varying fields. Consequently, the boundary conditions remain valid for time – varying fields, where a n is the unit normal vector to the boundary. E1t – E2t = 0 (or) (E1 – E2 ) X a n = 0 H1t – H2t = k (or) (H1 – H2 ) X a n = k D1n – D2n = ρs (or) (D1 – D2 ). an = ρ s B1n – B2n = 0 (or) (B1 – B2 ). a n = 0 However, for a perfect conductor (σ ≈∞) in a time – varying field, E = 0, H = 0 and J = 0 and hence, Bn = 0, E1t = 0 For a perfect dielectric (σ ≈ 0) the above equations hold except that k = 0.

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